Bohr Model Calculator

In 1913 Niels Bohr proposed that the electrons in a hydrogen atom could not orbit freely at any distance from the nucleus. Instead they can only exist on a series of discrete orbits, each with a particular fixed energy.

This idea made it possible to achieve something that earlier models could not: accurately predict the wavelengths of light emitted by hydrogen. Spectroscopists had measured this spectrum for over thirty years without being able to explain it.

This calculator treats the Bohr model in three ways: it calculates photons emitted or absorbed when an electron moves between energy levels, it fully describes a single orbit, and it converts measured pairs of energy levels into photon frequency and wavelength.

Bohr energy levels.

For a system with a single electron and nuclear charge Z, the energy of level n is given by:

Here n is the principal quantum number, a whole number that can take values of 1, 2, 3 and above. The constant E_R is the Rydberg energy. For hydrogen Z = 1 so the ground state energy is −13.6 eV.

The minus sign is important. The energy of a bound electron is less than that of a free electron, and the zero point is defined as being the state in which an electron is infinitely far from the nucleus. So all energy levels have negative values, tending gradually towards zero as n gets larger.

Because the energy is divided by the square of n, these energy levels are never evenly spaced. The transition from n = 1 to n = 2 requires 10.2 eV, while the transition from n = 2 to n = 3 only requires 1.89 eV. By the time you get to n = 10, the individual levels cluster together in a range less than one electron volt wide.

Formula for transitions (energy, wavelength, frequency)

Electrons change their energy levels by exchanging photons. The energy of a photon is exactly equal to the difference between two energy levels, no more and no less. This explains why atoms emit clear spectral lines rather than fuzzy bands of color.

If this energy is known, then the frequency and wavelength of the photon can be determined using Planck's relation.

Dividing by h and c gives the wave number. Opticians prefer the wave number because it is proportional to energy and a convenient number in inverse centimeters. In this form the Rydberg formula can be expressed, which was discovered decades before Bohr.

Bohr's actual merit is not in inventing this formula. Balmer and Rydberg had already found an approximation of this formula based on the measured wavelengths. What Bohr achieved was to derive this formula by quantizing the angular momentum of the electron, and explain why nature follows such orderly rules.

Example calculation (red line of hydrogen)

Consider a hydrogen atom with Z = 1 and an electron transition from n = 3 to n = 2. The two energy levels are as follows:

The energy difference is 1.88968 eV, so the photon also has that amount of energy. Converted to wavelength this results in:

This is a deep red spectral line found in hydrogen discharge tubes and the spectra of countless stars, it is called the Hydrogen-alpha line and is one of the most easily recognizable features across all of astronomy.

Example calculation (ultraviolet Lyman-alpha line)

For example, if an electron were to jump from energy state n = 2 to n = 1, then the difference in energy would be 13.60569 eV - 3.40142 eV, or 10.2043 eV, which is more than five times the energy of the red line.

The greater the energy, the shorter the wavelength. The wavelength of this photon is 121.50 nm and it falls in the deep ultraviolet region of the spectrum. Since air absorbs this, the Lyman-alpha line can only be observed from space or a vacuum chamber. For that reason, the Balmer lines were discovered first in visible light.

Spectral series

When the transitions to lower energy states are ordered, the hydrogen spectrum naturally breaks up into different series. Before it was known what connected these series, each series had been discovered by different people.

This is why the Balmer series is so important for hydrogen in astronomical observations; it is the only spectral series that lies in the visible region and can therefore be identified from starlight using normal optical telescopes.

Each spectral series has a limit. If one were to push the higher energy state infinitely far upwards, then the photon's energy would no longer increase but remain constant at the ionization energy for that particular lower energy state. Photons with an energy above this limit will not rip an electron out of a higher orbital; instead they will completely remove an electron from the atom. The excess energy is converted into the kinetic energy of the free electron.

The inside of a single track.

Bohr's quantization rule states that the angular momentum of an electron can only take on integer multiples of the reduced Planck constant. All other properties of the orbit are determined by this rule and Coulomb's law.

The Bohr radius a_0 is 52.918 pm and corresponds to the size of the hydrogen atom in its ground state. As the orbital radius is proportional to the square of n, an electron with n = 10 is located 100 times further from the nucleus than an electron with n = 1.

The speed decreases inversely with the value obtained by dividing 1 by n. In the ground state of hydrogen, the electron moves at a speed of 2187.7 km per second, which is 0.73 percent of the speed of light. This value is the fine structure constant. It's precisely because this value is small that the non-relativistic Bohr model works so well.

As the nuclear charge is increased, the atom shrinks and the speed of the electron increases. In uranium ions with only one remaining electron, the ground state velocity exceeds 60 percent of the speed of light. At this point relativistic effects dominate, and the Bohr model is no longer reliable.

The energy of the Coulomb orbit is also distributed in a particular ratio, which is called the virial theorem. The kinetic energy corresponds to the negative value of the total energy, while the potential energy is twice as large as the total energy.

Thus the ground state of hydrogen contains a kinetic energy of 13.6 eV and a potential energy of −27.2 eV, for a total energy of −13.6 eV. To free an electron, it takes 13.6 eV, which is the ionization energy of hydrogen.

Hydrogen-like ions (the true role of Z)

Bohr's formula can be applied to any system where only a single electron is orbiting the nucleus of an atom. He+, Li2+, and Be3+ all follow this formula with Z being the nuclear charge.

Since the energy is proportional to Z squared, this has a very large effect. For example, He+ has four times the ionization energy of hydrogen, so its ground state energy is -54.4 eV. In addition, the associated Balmer line moves from the visible region into the ultraviolet.

If there is a second electron present, the model breaks down. The energy of neutral helium is not four times that of hydrogen. This is because the two electrons repel each other and shield the nucleus, and this problem cannot be adequately corrected by adjusting Z. This is one limitation of the Bohr model and one reason why quantum mechanics had to be developed.

The reason why the values stated in textbooks do not match with the actually measured spectral lines.

When the Balmer-alpha line is calculated using a standard Rydberg energy of 13.6057 eV, it results in a value of 656.11 nm. An examination of spectroscopy tables shows that the measured value is 656.46 nm. Although the difference is small, it exists and cannot be attributed to rounding errors.

The reason for this is that the atomic nucleus is not fixed. As electrons and protons move around a common center of mass, the quantity to be used is not the electron's mass but its reduced mass.

Since the mass of a proton is about 1836 times that of an electron, the magnitude of this correction is only about 0.055 percent. When the correction option in this calculator is turned on, the hydrogen-alpha line shifts to 656.47 nm and the difference from the measured value is about a hundredth of a nanometre.

The larger the mass of the nucleus, the smaller the recoil and thus the correction. The correction is about half as large for deuterium and about a quarter as large for helium. This difference is measurable, and this was the method by which deuterium was discovered in 1931. Researchers found it from weak companion lines just next to the hydrogen spectral lines.

The scope of application of Bohr's model

The Bohr model derives the energy levels of hydrogen perfectly correctly. It is amazing that this picture which is ultimately physically incorrect works so well. Electrons do not move in circular orbits but instead exist as probability clouds with no defined paths.

This model can only be used for single electrons and does not explain finer structures within the individual energy levels. Actual spectral lines split into components with very small separations due to electron spin, relativistic corrections, and Lamb shifts, but these phenomena are completely ignored in the Bohr formula.

Also the model cannot explain how bright a spectral line is. To predict transition rates and selection rules, full quantum mechanics is required. So while Bohr's model tells us where spectral lines are, it does not tell us how strong they are.

Bohr himself bridged this gap with the so-called correspondence principle. The quantum theory must reproduce classical physics when the quantum numbers become very large. When n is very high, the individual energy levels are extremely close to each other and this group of levels becomes virtually continuous. This corresponds exactly to what classical physics predicts for a charge orbiting around the nucleus.

This calculator is for educational and reference purposes only. It calculates an idealized Bohr model for single-electron systems, with the option to apply mass corrections. Fine structure, hyperfine structure, Lamb shift, and other many-body effects are not included. For precision spectroscopic work, experimentally determined spectral lines from databases such as NIST Atomic Spectra Database should be used.